• Mix the flax egg and set aside.
• Combine the milk and butter in a microwave-safe bowl and heat for 1 minute. Set it aside for several minutes. Then stir in the flax egg and vinegar. The mixture should be between 105°F/40°C and 108°F/42°C (the same temperature as your finger if you dip it into the liquid).
• In a large bowl, add the flours, sugar, yeast, baking powder, and salt. Stir to combine. Pour this milk mixture into the flour mixture and stir together until a soft dough forms. Set aside.
  

For the Filling:

• Mix brown sugar, 1 tablespoon of gluten-free flour, and cinnamon.
  

Rolling the Dough:

• Once the dough is ready, cover a working area with a sheet of plastic wrap. Place the dough on top of the plastic wrap and cover it with another sheet. Use a rolling pin to roll the dough out to approximately 11x15 inches in size. It helps if you pull the top layer of plastic wrap off the dough and cover it again with the same sheet. It also helps to place an anti-slip pad to keep the dough from sliding as you roll it. 

Preparing Rolls:

• Spread softened butter across the dough. Sprinkle with cinnamon sugar filling mixture. Sprinkle this over the butter, covering all of it except for about a half-inch of the long edge furthest away from you.
  
• Starting from the long end closest to you, roll the dough tightly until you're left with one long roll.
  
• Spray an 8-inch round pan or pie pan with nonstick cooking spray. Cut into 9 evenly sized rolls about 1 ½ inches wide. Place rolls in the prepared pan. Cover with a kitchen towel, and set aside in a warm, draft-free area to rise until doubled in size, around 1 hour. See notes for options.
  
• When it's time to bake, preheat the oven to 350ᵒF/175°C. Place pan in a center rack and bake for about 24 to 26 minutes until the rolls are golden in color.

For the Icing:

• While the rolls are baking, prepare the icing. In a medium bowl, cream together the butter and vegan cream cheese until fluffy. Add the vanilla, followed by the powdered sugar beat until creamy. Stir in one tablespoon of the soy milk at a time to get a spreadable consistency.
• When the rolls have cooled slightly, remove them from the oven and add frosting immediately. Serve warm.
  
• Store cinnamon rolls in a sealed container at room temperature for up to 3 days or in the fridge for up to 7 days. These can be frozen for up to 2 months. To serve, cook in the microwave until heated through.